A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation

Abstract

Abstract:In this paper, we have presented and studied two types of the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation. We prove that the fractional order delay integral equation is Mittag-Leffler-Hyers-Ulam stable on a compact interval with respect to the Chebyshev and Bielecki norms by two notions.

On the impulsive implicit Ψ‐Hilfer fractional differential equations with delay

“…To examine the Ulam-Hyers-Mittag-Leffler (UHML) stability of the problem (1)-(2), we adopt the definitions given by Wang et al 18 …”

“…The existence and uniqueness of solutions and the Ulam-Hyers-Mittag-Leffler (UHML) stability of different kinds of fractional differential and integral equations with time delay have been investigated in Eghbali et al, 17 Wang et al, 18 and Niazi et al 19 by using Picard operator theory and abstract Gronwall lemma. Then again, there are many fascinating research papers involving Hilfer fractional derivative, which incorporates the Riemann-Liouville and Caputo fractional derivative as special cases.…”

“…13,15,16,[27][28][29][30][31] Further, looking towards the multidimensional utilization of impulsive FDES, 1,2 it is important to consider the new class implicit FDEs with impulse condition that incorporates a wide class of impulsive FDEs as particular cases. [16][17][18][19][20][21][22][23][24][25] The paper is organized as follows. If we take Ψ(t) = t and = 1, then the problem (1)-(4) reduces to implicit impulsive FDEs with the Caputo fractional derivative.…”

Stability via successive approximation for nonlinear implicit fractional differential equations

“…We mention here few recent works by Wang et al [11,12,13,14], Eghbali et al [15] and Wei et al [16]; also see the references cited therein.…”

On existence and stability results for nonlinear fractional delay differential equations

“…In recent years, some researchers extended the concept of Ulam type stabilities by using different techniques to various forms of fractional differential and fractional integral equations with different types of fractional derivative operators, for example, Wang and Xu [7] by applying the Laplace transform method have investigated the Hyers-Ulam stability of fractional linear differential equation with Riemann-Liouville fractional derivative, Eghbali and coauthers [8] proved that the fractional order delay integral equation is Mittag-Leffler-Hyers-Ulam stable on a compact interval with respect to the Chebyshev and Bielecki norms, Yu [9] studied β-Ulam-Hyers stability for a class of fractional differential equations with non-instantaneous impulses, Hyers-Ulam stability results for nonlinear fractional systems with coupled nonlocal initial conditions have been investigated in [10], Peng and Wang [11] discussed existence of solutions and Ulam-Hyers stability of Cauchy problem for nonlinear ordinary differential equations involving two Caputo fractional derivatives, Abbas in [12] dealt with existence, uniqueness and MittagLeffler-Ulam stablity of fractional integrodifferential equations.…”